- Open Access
Extensions of Banach contraction principle to partial cone metric spaces over a non-normal solid cone
© Jiang and Li; licensee Springer. 2013
- Received: 24 June 2013
- Accepted: 9 September 2013
- Published: 7 November 2013
In this paper, we present some extensions of Banach contraction principle to partial cone metric spaces over a non-normal solid cone, which improve many recent fixed point results in cone metric spaces and partial cone metric spaces. An example is given to support the usability of our results.
- Banach contraction principle
- partial cone metric space
- non-normal cone
- solid cone
The Banach contraction principle is the most celebrated fixed point theorem, which has been extended in various directions. In 2007, Huang and Zhang  introduced cone metric spaces and extended the Banach contraction principle to cone metric spaces over a normal solid cone, being unaware that cone metric spaces already existed under the name of K-metric spaces and K-normed spaces that were introduced and used in the middle of the 20th century in [2–9]. Furthermore, Huang and Zhang defined the convergence via interior points of the cone. Such an approach allows the investigation of the case that the cone is not necessarily normal, for example, the authors in [10–18] established many fixed point results and common fixed point results in cone metric spaces over a non-normal cone. In 2012, based on the definition of cone metric spaces and partial metric spaces, which were introduced by Matthews , Sonmez [20, 21] defined a partial cone metric space and considered the extensions of Banach contraction principle to partial cone metric spaces.
It is worth mentioning that in most of the preceding references concerned with fixed point results of contractions in cone metric spaces and partial cone metric spaces, the contractions are always assumed to be restricted with a constant. In , Agarwal considered a contraction restricted with a positive linear mapping and proved the following fixed point theorem in cone metric spaces.
Theorem 1 (See )
Then T has a unique fixed point .
It is clear that is a normal solid cone of endowed with the usual norm. Motivated by [10–18, 20, 21], we in this paper shall extend Theorem 1 to partial cone metric spaces over a non-normal solid cone of an abstract normed vector space.
for each and each , and
, where θ is the zero element of E.
Each cone P of E determines a partial order ⪯ on E by for each .
A cone P of a topological vector space E, is solid  if , where intP is the interior of P. For each with , we write . A cone P of a normed vector space , is normal  if there exists such that implies that for each , and the minimal N is called a normal constant of P.
Lemma 1 Let P be a solid cone of a normed vector space , and let be a sequence in E. Then implies that for each , there exists a positive integer such that , i.e., for all .
Proof For each , there exists some such that implies that for each . If , then for this ε, there exists a positive integer such that for each , and hence for each , i.e., for each . The proof is complete. □
Remark 1 The converse of Lemma 1 is true provided that P is normal. In fact, for each , there exists some such that , where N denotes the normal constant of P. Note that for this ϵ, there exists a positive integer such that for each , and so . Then for each by the normality of P. This forces that .
The following example shows that the converse of Lemma 1 may not be true if P is non-normal.
Example 1 Let with the norm and , which is a non-normal solid cone . Let . Clearly, , and so . On the other hand, let , then , and . By Lemma 1, for each , there exists a positive integer such that for all , and hence for all .
The pair is called a partial cone metric space over P.
Each cone metric is certainly a partial cone metric. The following example shows that there does exist some partial cone metric which is not a cone metric.
i.e., (p4) is satisfied for each . Hence p is partial cone metric, but not a cone metric, since for each with .
Each partial cone metric p on X over a solid cone generates a topology on X, which has a base of the family of open p-balls , where for each and each .
A sequence in X is θ-Cauchy, if for each , there exists a positive integer such that for all . The partial cone metric space is θ-complete, if each θ-Cauchy sequence of X converges to a point such that .
It follows from Lemma 1 and Remark 1 that each strongly convergent sequence of a partial cone metric space X is convergent whenever E is a normed vector space, and the converse is true provided that P is a normal. The following example will show that there exists some sequence of a partial cone metric, which is convergent but not strongly convergent if P is non-normal.
Example 3 Let , E and P be the same ones as those in Example 2, and let . Then , but . In fact, it is clear that , , and , where . Then by Lemma 1, for each , there exists a positive integer such that for all , i.e., . On the other hand, , and hence .
Definition 2 Let be a partial cone metric space over a solid cone P of a normed vector space . A sequence in X is Cauchy [20, 21], if there exists with such that . The partial cone metric space is complete [20, 21], if each Cauchy sequence of X strongly converges to a point such that .
If P is a normal solid cone of a normed vector space , then each complete partial cone metric space is θ-complete by Lemma 1 and Remark 1. But the converse is not true, the following example shows that a partial cone metric space which is θ-complete, is not necessarily complete.
Clearly, is a partial cone metric space, for each , , P is normal.
Let be a sequence in , where . If is θ-Cauchy, then by Remark 1 and the normality of P, , and so for each , there exists such that for each . Thus, for each and each . This means for each , i.e., . Therefore, , i.e., , and hence is θ-complete since .
Let for each n and each , and . It is clear that , and hence is a Cauchy sequence in . If there exists such that , then , which contradicts to the fact that since . This means for each , and so there does not exist such that . Therefore, is not complete.
In this section, we present some extensions of Banach contraction principle in the setting of partial cone metric spaces over a non-normal solid cone of an abstract normed vector space.
then there exists a positive integer such that , where is a subsequence of defined by .
which together with the arbitrary property of ϵ implies that , and hence by (p1) and (p3), i.e., is a fixed point of . Let be a fixed point of , i.e., . Note that by (4), then the inverse of exists, denote it by . Moreover by Neumann’s formula, . By (2), we have , and hence . Act it with , then . This implies that , and hence by (p1) and (p3). Hence is the unique fixed point of .
Note that is a fixed point of , then , i.e., is also a fixed point of . By the uniqueness of fixed point of , we have , i.e., is also a fixed point of T. Let y be a fixed point of T. It is clear that y is also a fixed point of , and so by the uniqueness of fixed point of . Hence is the unique fixed point of T. The proof is complete. □
Remark 2 It is clear that Theorem 1 is exactly a special case of Theorem 2 with and . Let for some constant , then , and so Theorem 1 of , Theorem 6 of  and Theorem 7 of  directly follow from Theorem 2. In addition, the normality of P necessarily assumed in [1, 9, 20, 21] has been removed in Theorem 2. Therefore, Theorem 2 indeed improves the corresponding results in [1, 9, 20, 21].
The following example shows the usability of Theorem 2.
Example 5 Let , E and P be the same ones as those in Example 2. Let for each , where , . Clearly, θ is the unique fixed point of T.
which implies that . Therefore by Gelfand’s formula, since , and hence T has a unique fixed point in X by Theorem 2.
However, the existence of fixed point of T cannot derive from the fixed point results in [1–21, 23], since P is non-normal, and p is not a cone metric by Example 1 and Example 2, and there does not exist a constant such that .
Then T has a unique fixed point , and for each , , where is defined by (3).
This forces that since , and so by (p1) and (p3). Hence is the unique fixed point of T. The proof is complete. □
Remark 3 Theorem 3 and Theorem 4 of  are special cases of Theorem 3 in the setting of cone metric spaces with , and , respectively, and Theorem 7 of  and Theorem 8 of  are special cases of Theorem 3 with , . In addition, P is not necessarily normal in Theorem 3. Compared with the corresponding results of [20, 21], the partial cone metric space X is only assumed to be θ-complete, but not complete in Theorem 2 and Theorem 3.
The work was supported by the Natural Science Foundation of China (11161022), the Natural Science Foundation of Jiangxi Province (20114BAB211006, 20122BAB201015), the Educational Department of Jiangxi Province (GJJ12280, GJJ13297) and Program for Excellent Youth Talents of JXUFE (201201).
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